Magnetic trajectories in Walker 3-manifolds (Q6635212)

This paper investigates magnetic trajectories in Walker 3-manifolds, a type of semi-Riemannian manifolds. The authors focus on geodesics, Killing magnetic trajectories, and contact magnetic trajectories within these manifolds. They also examine foliated Walker 3-manifolds, characterizing their properties and analyzing magnetic trajectories within this specific structure. Key results of the paper can be summarized as follows.\N\NProposition 3.1 provides explicit parametric equations for geodesics in strict Walker 3-manifolds, considering different cases based on the initial conditions and the value of \(\varepsilon\) which determines the signature of the metric. This result allows for a more detailed understanding of the paths particles would take in the absence of a magnetic field.\N\NTheorem 3.1 presents explicit formulas for magnetic trajectories under the influence of a uniform magnetic field, represented by the parallel vector field \(\partial t\), in strict Walker 3-manifolds. This theorem builds upon previous work by \textit{C.-L. Bejan} and \textit{S.-L. Druţă-Romaniuc} [Differ. Geom. Appl. 35, 106--116 (2014; Zbl 1322.53074)], offering a comprehensive description of these trajectories in a more general setting.\N\NIn contact Walker 3-manifolds, where the plane field \(\mathcal{N}^\bot\) forms a contact structure, the Reeb vector field \(\xi\) becomes a magnetic field. These trajectories are referred to as contact magnetic trajectories. Solving the resulting system of differential equations (4.2) is challenging, but explicit solutions have been found for certain functions \(f\), as demonstrated in Proposition 4.2.\N\NProposition 4.2 provides explicit expressions for spacelike, timelike, and lightlike contact magnetic trajectories in a specific type of contact Walker 3-manifold where the metric function \(f\) has a particular form, \(f(t,x,y)=a g(y) e^{cx+t}\). This result demonstrates the possibility of obtaining explicit solutions for contact magnetic trajectories under certain conditions.\N\NFoliated Walker 3-manifolds are characterized by the integrability of the plane field \(\mathcal{N}^\bot\), indicating that \(\mathcal{N}^\bot\) is formed by a family of surfaces. This foliation influences the magnetic trajectories associated with the magnetic field \(\mathcal{N}\). In these manifolds, \(\mathcal{N}^\bot\) cannot be a contact structure, and alternative conditions like \(d\eta=0\) need to be considered.\N\NProposition 5.1 characterizes magnetic trajectories in foliated Walker 3-manifolds, where the plane field \(\mathcal{N}^\bot\) is integrable but not contact. This result highlights the distinct behavior of magnetic trajectories in these manifolds compared to those with contact structures.\N\NIn conclusion, this paper shows that the characteristics of Walker 3-manifolds, such as their strictness, contact structure, and foliation, directly impact the behavior of magnetic trajectories. The specific form of the metric function \(f\) and the properties of the magnetic field further shape the trajectories. The presented results on magnetic trajectories in foliated Walker 3-manifolds, particularly the non-contact ones, in contrast to previous studies that primarily examined trajectories in contact manifolds, provide new insights into the research on magnetic curves of 3-manifolds.